Approximate Irrationals

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Answer Key & Explanations

Approximate Irrationals · Grade 8 · Worksheet 3

1. The area of a square garden is 150 square meters. Between which two consecutive whole numbers does the length of one side of the garden lie? Answer: 12 Solution: We are told the area of the square garden is 150 square meters. Recall the formula for the area of a square. Area = side × side, or A = s².
   Full step-by-step solution

   We are told the area of the square garden is 150 square meters. Step 1: Recall the formula for the area of a square. Area = side × side, or A = s². Step 2: Substitute the given area into the formula. s² = 150 Step 3: Find the side length by taking the square root of both sides. s = √150 Step 4: Simplify √150 if possible. √150 = √(25 × 6) = √25 × √6 = 5 × √6 Step 5: Estimate √6. We know 2² = 4 and 3² = 9, so √6 is between 2 and 3. More precisely: 2.4² = 5.76 2.45² = 6.0025 (very close to 6) So √6 ≈ 2.45 Step 6: Multiply by 5. s ≈ 5 × 2.45 = 12.25 Step 7: Determine between which two consecutive whole numbers 12.25 lies. 12 < 12.25 < 13 So the side length lies between 12 and 13. Final answer: 12

2. Liam is designing a circular garden with a diameter of 12 meters. He needs to calculate the area to determine how much soil to purchase. Using π ≈ 3.14, what is the approximate area of the garden in square meters? Answer: 113.04 Solution: Recall the formula for the area of a circle. A = π × r² where r is the radius and π is approximately 3.14. Find the radius of the circle.
   Full step-by-step solution

   Step 1: Recall the formula for the area of a circle. The area A of a circle is given by: A = π × r² where r is the radius and π is approximately 3.14. Step 2: Find the radius of the circle. The problem gives the diameter as 12 meters. The radius is half of the diameter: r = diameter / 2 = 12 / 2 = 6 meters. Step 3: Substitute the values into the formula. A = 3.14 × (6)² Step 4: Calculate the square of the radius. (6)² = 6 × 6 = 36 Step 5: Multiply by π. A = 3.14 × 36 Step 6: Perform the multiplication. First: 3 × 36 = 108 Then: 0.14 × 36 = 5.04 Add them: 108 + 5.04 = 113.04 Step 7: State the final answer. The approximate area of the garden is 113.04 square meters.

3. √(86) ≈ ? (to the nearest tenth) Answer: 9.3 Solution: Identify perfect squares near 86. 9^2 = 81 and 10^2 = 100. So √86 is between 9 and 10.
   Full step-by-step solution

   Step 1: Identify perfect squares near 86. 9^2 = 81 and 10^2 = 100. So √86 is between 9 and 10. Step 2: Since 86 is closer to 81 than to 100, the square root is closer to 9. The difference from 81 is 5, and from 100 is 14. Step 3: Try 9.3: 9.3 × 9.3 = 86.49. This is 0.49 above 86. Step 4: Try 9.2: 9.2 × 9.2 = 84.64. This is 1.36 below 86. Step 5: Since 9.3^2 = 86.49 is only 0.49 above 86, and 9.2^2 = 84.64 is 1.36 below, 9.3 is the better approximation to the nearest tenth. The answer is 9.3.

4. Emma is designing a rectangular garden for her school's environmental project. The garden's length is √72 feet and its width is √32 feet. She needs to calculate the approximate area to determine how much mulch to purchase. What is the approximate area of Emma's garden in square feet, rounded to the nearest whole number? Answer: 48 Solution: The area of a rectangle is length × width, so Area = √72 × √32 Combine the square roots: √72 × √32 = √(72 × 32) Multiply the numbers inside the square root: 72 × 32 = 2304 Simplify the square root: √2304 = 48 Since 48 is already a whole number, the approximate area is 48 square feet.
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width, so Area = √72 × √32 Step 2: Combine the square roots: √72 × √32 = √(72 × 32) Step 3: Multiply the numbers inside the square root: 72 × 32 = 2304 Step 4: Simplify the square root: √2304 = 48 Step 5: Since 48 is already a whole number, the approximate area is 48 square feet. The answer is 48.

5. Hana is creating a square-shaped garden in her backyard. She wants the area of the garden to be exactly 85 square meters so she can plant a specific number of vegetables. To buy fencing for the perimeter, she needs to know the side length of the garden. Since 85 is not a perfect square, she needs to approximate the side length to the nearest tenth of a meter. What is the approximate side length of Hana's garden, rounded to the nearest tenth of a meter? Answer: 9.2 Solution: Find the two perfect squares closest to 85. 9^2 = 81 and 10^2 = 100. So, sqrt(85) is between 9 and 10.
   Full step-by-step solution

   Step 1: Find the two perfect squares closest to 85. 9^2 = 81 and 10^2 = 100. So, sqrt(85) is between 9 and 10. Step 2: Try 9.2. 9.2^2 = 84.64, which is less than 85. Step 3: Try 9.3. 9.3^2 = 86.49, which is greater than 85. Step 4: Since 84.64 is closer to 85 than 86.49 is (85 - 84.64 = 0.36, 86.49 - 85 = 1.49), sqrt(85) is closer to 9.2. Step 5: Therefore, the approximate side length is 9.2 meters.

6. √(2) × √(8) = ? Answer: 4 Solution: Write down the original problem. √(2) × √(8) Use the multiplication rule for square roots. The rule says: √(a) × √(b) = √(a × b) So: √(2) × √(8) = √(2 × 8) Multiply the numbers inside the square root.
   Full step-by-step solution

   Step 1: Write down the original problem. We have: √(2) × √(8) Step 2: Use the multiplication rule for square roots. The rule says: √(a) × √(b) = √(a × b) So: √(2) × √(8) = √(2 × 8) Step 3: Multiply the numbers inside the square root. 2 × 8 = 16 So we have: √(16) Step 4: Simplify √(16). We know 16 = 4 × 4, so √(16) = 4. Step 5: State the final answer. Therefore, √(2) × √(8) = 4.